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Non-Universal Moderate Deviation Principle for the Nodal Length of Arithmetic Random Waves

Claudio Macci, Maurizia Rossi, Anna Vidotto

Abstract

Inspired by the recent work [MRT21], we prove a non-universal non-central Moderate Deviation principle for the nodal length of arithmetic random waves (Gaussian Laplace eigenfunctions on the standard flat torus) both on the whole manifold and on shrinking toral domains. Second order fluctuations for the latter were established in [MPRW16] and [BMW20] respectively, by means of chaotic expansions, number theoretical estimates and full correlation phenomena. Our proof is simple and relies on the interplay between the long memory behavior of arithmetic random waves and the chaotic expansion of the nodal length, as well as on well-known techniques in Large Deviation theory (the contraction principle and the concept of exponential equivalence).

Non-Universal Moderate Deviation Principle for the Nodal Length of Arithmetic Random Waves

Abstract

Inspired by the recent work [MRT21], we prove a non-universal non-central Moderate Deviation principle for the nodal length of arithmetic random waves (Gaussian Laplace eigenfunctions on the standard flat torus) both on the whole manifold and on shrinking toral domains. Second order fluctuations for the latter were established in [MPRW16] and [BMW20] respectively, by means of chaotic expansions, number theoretical estimates and full correlation phenomena. Our proof is simple and relies on the interplay between the long memory behavior of arithmetic random waves and the chaotic expansion of the nodal length, as well as on well-known techniques in Large Deviation theory (the contraction principle and the concept of exponential equivalence).
Paper Structure (22 sections, 14 theorems, 133 equations)

This paper contains 22 sections, 14 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Plan of the paper.
  3. Background and notation
  4. Arithmetic random waves
  5. Nodal length: prior work
  6. Mean and variance
  7. Asymptotic distribution
  8. Shrinking domains
  9. Large Deviation Principles
  10. Main Results
  11. On the proof of the main results
  12. The spherical case
  13. Chaotic expansions
  14. Wiener chaos
  15. Explicit formulas
  16. ...and 7 more sections

Key Result

Theorem 2.2

Let $\left\lbrace n_{j} \right\rbrace_j\subseteq S$ be a subsequence of $S$ satisfying $\mathcal{N}_{n_{j}}\rightarrow\infty$, such that the sequence $\left\lbrace|\widehat{\mu}_{n_j}(4)|\right\rbrace_j$ converges, that is: for some $\eta \in [0,1]$. Then where $\mathcal{M}_\eta$ is defined as in e:r.

Theorems & Definitions (19)

  • Definition 2.1
  • Theorem 2.2: Theorem 1.1, MPRW:16
  • Theorem 2.3: Theorem 2, PR:18
  • Theorem 2.4: Theorem 1.1, BMW:20
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 9 more